Seminar in General Relativity

Summer 2007

Instructor: Joshua Reyes
Time: M Th 5:00PM - 6:30PM
Start Date: 4 June 2007
Location: Taffee Tanimoto Conference Room (S-3-180)

Recommended Reading:

The graphic for the poster came from a comic by Kevin de Leplante called How to Build Your Own Wormhole (Suitable for Interstellar Travel). In fact, the comic inspired the layout and title for the poster, too. Almost nothing I do is ever my own.

Tenative syllabus [PDF]. Poster [PDF].

If there is something that excites you about differential geometry or general relativity, I'll try to work it into the seminar (to the best of my abilities). Please do not hesitate to send me an email with any questions, comments, concerns, anecdotes, or other short stories.

You should not feel like you have to finish all the reading assignments. Each of the sources approaches the material in wildly different ways and levels of rigor. Try to find a source that suits you. And let me know what you think of the texts. I'm not following any one source all that closely (though if I had to pick one, I guess it'd be O'Neill), so look around.

Lecture Date Exercises Reading
1 4 June Draft [PDF] [DVI]
7 June No Class.
2 11 June Review the derivative from multivariable calculus. In particular focus on the chain rule and product rule. To get you going, here's something to think about. Consider the map T on Mat(n, n), the space of n×n matrices with real coefficients, taking a matrix A to its square A2. What is the derivative D(T(A)) when n = 1? What is it when n > 1? Hubbard and Hubbard chapter 1
3 14 June Show that any open set of a manifold is, itself, a manifold by constructing an altas for it explicitly.
4 18 June Tangent vectors, vector fields, differentials. Show that differential dxi(V) does, indeed, select the i-th coordinate from the vector field V. Show that the dxi form a dual basis to the partial derivatives ∂i according to the definitions. That is, dxi(∂j) is 1 if i=j and 0 otherwise.
5 21 June Look up the statement of the inverse function theorem, peak at a proof if you feel like it. Read about submanifolds, especially submersions. For those who want to put everything in Rn, check out Whitney's embedding theorem. Guillemin and Pollack, Chapter 1.3–1.4
6 25 June Review multilinear algebra: in particular, know what a module is. O'Neill, first half of Chapter 2; Boothby, Chapter 5; Wald Chapter 2.2–2.4; Chapter 5 in d'Inverno could be helpful.
7 28 June Derive the transformation law for one-forms using the coordinate expression for the differential of a smooth map between manifolds and the definition of a dual basis. Use the definition of the tensor product to derive the transformation law for a general tensor of type (r, s). Boothby, Chapter 7.1–7.4; O'Neill, second half of Chapter 3 (scalar products), an assortment from Chapter 4; Wald 3.1–3.3; d'Inverno, some of Chapter 6
8 2 July Show that the set of isometries on a semi-Riemannian manifold (M, g) forms a group; i.e., the identity map is an isometry, the composition of two isometries is an isometry, and the inverse of an isometry is an isometry. (Recall: a smooth map φ: MM is an isometry if it is a diffeomorphism such that φ*(g) = g.)
5 July No class.
9 9 July For a null vector n and another vector w that is not orthogonal to n, show that the operator h = g + wn/⟨w, n⟩ is a projection operator, but is not symmetric. I.e., show that i) h(n) = 0, ii) h(v) = v if and only if ⟨v, n⟩ = 0, iii) that h2 = h(h) = h, and iv) tr h = dim M-1.
(Hint: Use local coordinates: hαβ = δα&beta+wα nβ/(∑ wμnμ) Justify why you can do the calculations locally.)
10 12 July Define the curl of a vector field V by its action on two other vector fields X and Y, namely, curl V(X, Y) = ⟨DX V, Y⟩ − ⟨DY V, X⟩.
Show that curl V is an anti-symmetric (0, 2)-tensor. (Such a creature is called a 2-form.) Find its components relative to a local coordinate chart. (Hint: Express V in local coordinates and let X = ∂i and Y = ∂j.)
11 16 July Let D be a tensor derivation. Let its action on the basis vectors in a coordinate chart be D(∂i) = ∑ Fjij. Show that for the one-form dual basis that D(dxj) = −∑ Fji dxi. (Notice that the indices i and j have been swapped.) (Hint: Using the axioms of a tensor derivation, apply the derivation to the expression θ(X) = C(θX), where θ is a one-form, X is a vector field, and C is a contraction map, to find the action of a derivation on a one-form in terms of its action on vector fields and functions.)
12 19 July Let D be the Levi-Civita covariant derivative on a semi-Riemannian manifold (M, g). Show that the metric tensor g is parallel with respect to D; i.e., DV g = 0 for all vector fields V. (Note: this is equivalent to axiom of metric compatability presented in lecture.)
13 23 July 1. Prove that if a geodesic is timelike at a given point p, then it is timelike everywhere along its length. Prove similar results for null and spacelike geodesics.
2. For the 2-dimensional metric ds2 = (dx2 − dt2)/t2, find all the Christoffel symbols Γijk [only four of them are non-zero], and find all the timelike geodesic curves.
14 26 July Calculate the Riemann curvature tensor for a semi-Riemannian manifold (M, g) for the cases dim M = 1, 2, or 3.
15 30 July 1. Let G = Ric − S g/2 be the Einstein curvature tensor, Ric be the Ricci curvature tensor, S be scalar curvature, and g is the metric tensor of a 4-dimensional semi-Riemannian manifold. Prove that Ric = G − tr(G) g/2.
2. A Riemannian manifold (M, g) is called Einstein if the Ricci curvature of M is a scalar multiple of the metric, Ric = cg, for some constant c. Show that for dim M ≥ 3, Ric = λg for some function λ implies the manifold is indeed Einstein. (Hint: Use divergence to get one constraint on λ. Apply contraction to get another.)
16 2 August 1. Show that [X, Y] = DX Y − DY X completes our proof that the Hessian of f, H f(X, Y) = D(Df)(X, Y) = XYf − DX Yf, is symmetric. (Recall: We left it at H f = YXf − DY Xf.) Demonstrate C-linearity in Y to prove that it is a (0, 2)-tensor. (Why only in Y?) Go on to show that H f(X, Y) = ⟨DX grad f, Y⟩. (Hint: Remember that grad f is the vector that is metrically equivalent to df. Combine that fact and compatibility of the covariant derivative with the metric to prove the result.)
2. Define the curl of a vector field V as in Lecture 10 above. Show that curl (grad f) = 0.
17 6 August Prove (from the definition) that the wedge product is distributive. Namely, show that (cα + β)∧&gamma = c(α∧γ) + β∧&gamma, for any constant c and differential forms α, β, and γ
18 9 August 1. Prove that vectors v1,…,vk are linearly independent if and only if v1∧⋅⋅⋅∧vk ≠ 0. (Hint: You may find it useful to extend the set of vectors to a basis.)
2. Let vV be a non-zero vector, and let w ∈ ∧k V*. Prove that vw = 0 if and only if there exists u ∈ ∧k − 1 V* such that w = vu.